Abstract: The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

29.
O.Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees and , Lecture Notes in Math. 1060 (1984), 187-200, see also a preprint with a more modern and detailed presentation available at http://www.math.uu.se/oleg/preprints.html. MR 0770238 (87i:14029)

O.Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees and , Lecture Notes in Math. 1060 (1984), 187-200, see also a preprint with a more modern and detailed presentation available at http://www.math.uu.se/oleg/preprints.html. MR 0770238 (87i:14029)